1*> \brief <b> ZGEEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices</b> 2* 3* =========== DOCUMENTATION =========== 4* 5* Online html documentation available at 6* http://www.netlib.org/lapack/explore-html/ 7* 8*> \htmlonly 9*> Download ZGEEVX + dependencies 10*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zgeevx.f"> 11*> [TGZ]</a> 12*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zgeevx.f"> 13*> [ZIP]</a> 14*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zgeevx.f"> 15*> [TXT]</a> 16*> \endhtmlonly 17* 18* Definition: 19* =========== 20* 21* SUBROUTINE ZGEEVX( BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, W, VL, 22* LDVL, VR, LDVR, ILO, IHI, SCALE, ABNRM, RCONDE, 23* RCONDV, WORK, LWORK, RWORK, INFO ) 24* 25* .. Scalar Arguments .. 26* CHARACTER BALANC, JOBVL, JOBVR, SENSE 27* INTEGER IHI, ILO, INFO, LDA, LDVL, LDVR, LWORK, N 28* DOUBLE PRECISION ABNRM 29* .. 30* .. Array Arguments .. 31* DOUBLE PRECISION RCONDE( * ), RCONDV( * ), RWORK( * ), 32* $ SCALE( * ) 33* COMPLEX*16 A( LDA, * ), VL( LDVL, * ), VR( LDVR, * ), 34* $ W( * ), WORK( * ) 35* .. 36* 37* 38*> \par Purpose: 39* ============= 40*> 41*> \verbatim 42*> 43*> ZGEEVX computes for an N-by-N complex nonsymmetric matrix A, the 44*> eigenvalues and, optionally, the left and/or right eigenvectors. 45*> 46*> Optionally also, it computes a balancing transformation to improve 47*> the conditioning of the eigenvalues and eigenvectors (ILO, IHI, 48*> SCALE, and ABNRM), reciprocal condition numbers for the eigenvalues 49*> (RCONDE), and reciprocal condition numbers for the right 50*> eigenvectors (RCONDV). 51*> 52*> The right eigenvector v(j) of A satisfies 53*> A * v(j) = lambda(j) * v(j) 54*> where lambda(j) is its eigenvalue. 55*> The left eigenvector u(j) of A satisfies 56*> u(j)**H * A = lambda(j) * u(j)**H 57*> where u(j)**H denotes the conjugate transpose of u(j). 58*> 59*> The computed eigenvectors are normalized to have Euclidean norm 60*> equal to 1 and largest component real. 61*> 62*> Balancing a matrix means permuting the rows and columns to make it 63*> more nearly upper triangular, and applying a diagonal similarity 64*> transformation D * A * D**(-1), where D is a diagonal matrix, to 65*> make its rows and columns closer in norm and the condition numbers 66*> of its eigenvalues and eigenvectors smaller. The computed 67*> reciprocal condition numbers correspond to the balanced matrix. 68*> Permuting rows and columns will not change the condition numbers 69*> (in exact arithmetic) but diagonal scaling will. For further 70*> explanation of balancing, see section 4.10.2 of the LAPACK 71*> Users' Guide. 72*> \endverbatim 73* 74* Arguments: 75* ========== 76* 77*> \param[in] BALANC 78*> \verbatim 79*> BALANC is CHARACTER*1 80*> Indicates how the input matrix should be diagonally scaled 81*> and/or permuted to improve the conditioning of its 82*> eigenvalues. 83*> = 'N': Do not diagonally scale or permute; 84*> = 'P': Perform permutations to make the matrix more nearly 85*> upper triangular. Do not diagonally scale; 86*> = 'S': Diagonally scale the matrix, ie. replace A by 87*> D*A*D**(-1), where D is a diagonal matrix chosen 88*> to make the rows and columns of A more equal in 89*> norm. Do not permute; 90*> = 'B': Both diagonally scale and permute A. 91*> 92*> Computed reciprocal condition numbers will be for the matrix 93*> after balancing and/or permuting. Permuting does not change 94*> condition numbers (in exact arithmetic), but balancing does. 95*> \endverbatim 96*> 97*> \param[in] JOBVL 98*> \verbatim 99*> JOBVL is CHARACTER*1 100*> = 'N': left eigenvectors of A are not computed; 101*> = 'V': left eigenvectors of A are computed. 102*> If SENSE = 'E' or 'B', JOBVL must = 'V'. 103*> \endverbatim 104*> 105*> \param[in] JOBVR 106*> \verbatim 107*> JOBVR is CHARACTER*1 108*> = 'N': right eigenvectors of A are not computed; 109*> = 'V': right eigenvectors of A are computed. 110*> If SENSE = 'E' or 'B', JOBVR must = 'V'. 111*> \endverbatim 112*> 113*> \param[in] SENSE 114*> \verbatim 115*> SENSE is CHARACTER*1 116*> Determines which reciprocal condition numbers are computed. 117*> = 'N': None are computed; 118*> = 'E': Computed for eigenvalues only; 119*> = 'V': Computed for right eigenvectors only; 120*> = 'B': Computed for eigenvalues and right eigenvectors. 121*> 122*> If SENSE = 'E' or 'B', both left and right eigenvectors 123*> must also be computed (JOBVL = 'V' and JOBVR = 'V'). 124*> \endverbatim 125*> 126*> \param[in] N 127*> \verbatim 128*> N is INTEGER 129*> The order of the matrix A. N >= 0. 130*> \endverbatim 131*> 132*> \param[in,out] A 133*> \verbatim 134*> A is COMPLEX*16 array, dimension (LDA,N) 135*> On entry, the N-by-N matrix A. 136*> On exit, A has been overwritten. If JOBVL = 'V' or 137*> JOBVR = 'V', A contains the Schur form of the balanced 138*> version of the matrix A. 139*> \endverbatim 140*> 141*> \param[in] LDA 142*> \verbatim 143*> LDA is INTEGER 144*> The leading dimension of the array A. LDA >= max(1,N). 145*> \endverbatim 146*> 147*> \param[out] W 148*> \verbatim 149*> W is COMPLEX*16 array, dimension (N) 150*> W contains the computed eigenvalues. 151*> \endverbatim 152*> 153*> \param[out] VL 154*> \verbatim 155*> VL is COMPLEX*16 array, dimension (LDVL,N) 156*> If JOBVL = 'V', the left eigenvectors u(j) are stored one 157*> after another in the columns of VL, in the same order 158*> as their eigenvalues. 159*> If JOBVL = 'N', VL is not referenced. 160*> u(j) = VL(:,j), the j-th column of VL. 161*> \endverbatim 162*> 163*> \param[in] LDVL 164*> \verbatim 165*> LDVL is INTEGER 166*> The leading dimension of the array VL. LDVL >= 1; if 167*> JOBVL = 'V', LDVL >= N. 168*> \endverbatim 169*> 170*> \param[out] VR 171*> \verbatim 172*> VR is COMPLEX*16 array, dimension (LDVR,N) 173*> If JOBVR = 'V', the right eigenvectors v(j) are stored one 174*> after another in the columns of VR, in the same order 175*> as their eigenvalues. 176*> If JOBVR = 'N', VR is not referenced. 177*> v(j) = VR(:,j), the j-th column of VR. 178*> \endverbatim 179*> 180*> \param[in] LDVR 181*> \verbatim 182*> LDVR is INTEGER 183*> The leading dimension of the array VR. LDVR >= 1; if 184*> JOBVR = 'V', LDVR >= N. 185*> \endverbatim 186*> 187*> \param[out] ILO 188*> \verbatim 189*> ILO is INTEGER 190*> \endverbatim 191*> 192*> \param[out] IHI 193*> \verbatim 194*> IHI is INTEGER 195*> ILO and IHI are integer values determined when A was 196*> balanced. The balanced A(i,j) = 0 if I > J and 197*> J = 1,...,ILO-1 or I = IHI+1,...,N. 198*> \endverbatim 199*> 200*> \param[out] SCALE 201*> \verbatim 202*> SCALE is DOUBLE PRECISION array, dimension (N) 203*> Details of the permutations and scaling factors applied 204*> when balancing A. If P(j) is the index of the row and column 205*> interchanged with row and column j, and D(j) is the scaling 206*> factor applied to row and column j, then 207*> SCALE(J) = P(J), for J = 1,...,ILO-1 208*> = D(J), for J = ILO,...,IHI 209*> = P(J) for J = IHI+1,...,N. 210*> The order in which the interchanges are made is N to IHI+1, 211*> then 1 to ILO-1. 212*> \endverbatim 213*> 214*> \param[out] ABNRM 215*> \verbatim 216*> ABNRM is DOUBLE PRECISION 217*> The one-norm of the balanced matrix (the maximum 218*> of the sum of absolute values of elements of any column). 219*> \endverbatim 220*> 221*> \param[out] RCONDE 222*> \verbatim 223*> RCONDE is DOUBLE PRECISION array, dimension (N) 224*> RCONDE(j) is the reciprocal condition number of the j-th 225*> eigenvalue. 226*> \endverbatim 227*> 228*> \param[out] RCONDV 229*> \verbatim 230*> RCONDV is DOUBLE PRECISION array, dimension (N) 231*> RCONDV(j) is the reciprocal condition number of the j-th 232*> right eigenvector. 233*> \endverbatim 234*> 235*> \param[out] WORK 236*> \verbatim 237*> WORK is COMPLEX*16 array, dimension (MAX(1,LWORK)) 238*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK. 239*> \endverbatim 240*> 241*> \param[in] LWORK 242*> \verbatim 243*> LWORK is INTEGER 244*> The dimension of the array WORK. If SENSE = 'N' or 'E', 245*> LWORK >= max(1,2*N), and if SENSE = 'V' or 'B', 246*> LWORK >= N*N+2*N. 247*> For good performance, LWORK must generally be larger. 248*> 249*> If LWORK = -1, then a workspace query is assumed; the routine 250*> only calculates the optimal size of the WORK array, returns 251*> this value as the first entry of the WORK array, and no error 252*> message related to LWORK is issued by XERBLA. 253*> \endverbatim 254*> 255*> \param[out] RWORK 256*> \verbatim 257*> RWORK is DOUBLE PRECISION array, dimension (2*N) 258*> \endverbatim 259*> 260*> \param[out] INFO 261*> \verbatim 262*> INFO is INTEGER 263*> = 0: successful exit 264*> < 0: if INFO = -i, the i-th argument had an illegal value. 265*> > 0: if INFO = i, the QR algorithm failed to compute all the 266*> eigenvalues, and no eigenvectors or condition numbers 267*> have been computed; elements 1:ILO-1 and i+1:N of W 268*> contain eigenvalues which have converged. 269*> \endverbatim 270* 271* Authors: 272* ======== 273* 274*> \author Univ. of Tennessee 275*> \author Univ. of California Berkeley 276*> \author Univ. of Colorado Denver 277*> \author NAG Ltd. 278* 279*> \date June 2016 280* 281* @precisions fortran z -> c 282* 283*> \ingroup complex16GEeigen 284* 285* ===================================================================== 286 SUBROUTINE ZGEEVX( BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, W, VL, 287 $ LDVL, VR, LDVR, ILO, IHI, SCALE, ABNRM, RCONDE, 288 $ RCONDV, WORK, LWORK, RWORK, INFO ) 289 implicit none 290* 291* -- LAPACK driver routine (version 3.7.0) -- 292* -- LAPACK is a software package provided by Univ. of Tennessee, -- 293* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 294* June 2016 295* 296* .. Scalar Arguments .. 297 CHARACTER BALANC, JOBVL, JOBVR, SENSE 298 INTEGER IHI, ILO, INFO, LDA, LDVL, LDVR, LWORK, N 299 DOUBLE PRECISION ABNRM 300* .. 301* .. Array Arguments .. 302 DOUBLE PRECISION RCONDE( * ), RCONDV( * ), RWORK( * ), 303 $ SCALE( * ) 304 COMPLEX*16 A( LDA, * ), VL( LDVL, * ), VR( LDVR, * ), 305 $ W( * ), WORK( * ) 306* .. 307* 308* ===================================================================== 309* 310* .. Parameters .. 311 DOUBLE PRECISION ZERO, ONE 312 PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 ) 313* .. 314* .. Local Scalars .. 315 LOGICAL LQUERY, SCALEA, WANTVL, WANTVR, WNTSNB, WNTSNE, 316 $ WNTSNN, WNTSNV 317 CHARACTER JOB, SIDE 318 INTEGER HSWORK, I, ICOND, IERR, ITAU, IWRK, K, 319 $ LWORK_TREVC, MAXWRK, MINWRK, NOUT 320 DOUBLE PRECISION ANRM, BIGNUM, CSCALE, EPS, SCL, SMLNUM 321 COMPLEX*16 TMP 322* .. 323* .. Local Arrays .. 324 LOGICAL SELECT( 1 ) 325 DOUBLE PRECISION DUM( 1 ) 326* .. 327* .. External Subroutines .. 328 EXTERNAL DLABAD, DLASCL, XERBLA, ZDSCAL, ZGEBAK, ZGEBAL, 329 $ ZGEHRD, ZHSEQR, ZLACPY, ZLASCL, ZSCAL, ZTREVC3, 330 $ ZTRSNA, ZUNGHR 331* .. 332* .. External Functions .. 333 LOGICAL LSAME 334 INTEGER IDAMAX, ILAENV 335 DOUBLE PRECISION DLAMCH, DZNRM2, ZLANGE 336 EXTERNAL LSAME, IDAMAX, ILAENV, DLAMCH, DZNRM2, ZLANGE 337* .. 338* .. Intrinsic Functions .. 339 INTRINSIC DBLE, DCMPLX, CONJG, AIMAG, MAX, SQRT 340* .. 341* .. Executable Statements .. 342* 343* Test the input arguments 344* 345 INFO = 0 346 LQUERY = ( LWORK.EQ.-1 ) 347 WANTVL = LSAME( JOBVL, 'V' ) 348 WANTVR = LSAME( JOBVR, 'V' ) 349 WNTSNN = LSAME( SENSE, 'N' ) 350 WNTSNE = LSAME( SENSE, 'E' ) 351 WNTSNV = LSAME( SENSE, 'V' ) 352 WNTSNB = LSAME( SENSE, 'B' ) 353 IF( .NOT.( LSAME( BALANC, 'N' ) .OR. LSAME( BALANC, 'S' ) .OR. 354 $ LSAME( BALANC, 'P' ) .OR. LSAME( BALANC, 'B' ) ) ) THEN 355 INFO = -1 356 ELSE IF( ( .NOT.WANTVL ) .AND. ( .NOT.LSAME( JOBVL, 'N' ) ) ) THEN 357 INFO = -2 358 ELSE IF( ( .NOT.WANTVR ) .AND. ( .NOT.LSAME( JOBVR, 'N' ) ) ) THEN 359 INFO = -3 360 ELSE IF( .NOT.( WNTSNN .OR. WNTSNE .OR. WNTSNB .OR. WNTSNV ) .OR. 361 $ ( ( WNTSNE .OR. WNTSNB ) .AND. .NOT.( WANTVL .AND. 362 $ WANTVR ) ) ) THEN 363 INFO = -4 364 ELSE IF( N.LT.0 ) THEN 365 INFO = -5 366 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN 367 INFO = -7 368 ELSE IF( LDVL.LT.1 .OR. ( WANTVL .AND. LDVL.LT.N ) ) THEN 369 INFO = -10 370 ELSE IF( LDVR.LT.1 .OR. ( WANTVR .AND. LDVR.LT.N ) ) THEN 371 INFO = -12 372 END IF 373* 374* Compute workspace 375* (Note: Comments in the code beginning "Workspace:" describe the 376* minimal amount of workspace needed at that point in the code, 377* as well as the preferred amount for good performance. 378* CWorkspace refers to complex workspace, and RWorkspace to real 379* workspace. NB refers to the optimal block size for the 380* immediately following subroutine, as returned by ILAENV. 381* HSWORK refers to the workspace preferred by ZHSEQR, as 382* calculated below. HSWORK is computed assuming ILO=1 and IHI=N, 383* the worst case.) 384* 385 IF( INFO.EQ.0 ) THEN 386 IF( N.EQ.0 ) THEN 387 MINWRK = 1 388 MAXWRK = 1 389 ELSE 390 MAXWRK = N + N*ILAENV( 1, 'ZGEHRD', ' ', N, 1, N, 0 ) 391* 392 IF( WANTVL ) THEN 393 CALL ZTREVC3( 'L', 'B', SELECT, N, A, LDA, 394 $ VL, LDVL, VR, LDVR, 395 $ N, NOUT, WORK, -1, RWORK, -1, IERR ) 396 LWORK_TREVC = INT( WORK(1) ) 397 MAXWRK = MAX( MAXWRK, LWORK_TREVC ) 398 CALL ZHSEQR( 'S', 'V', N, 1, N, A, LDA, W, VL, LDVL, 399 $ WORK, -1, INFO ) 400 ELSE IF( WANTVR ) THEN 401 CALL ZTREVC3( 'R', 'B', SELECT, N, A, LDA, 402 $ VL, LDVL, VR, LDVR, 403 $ N, NOUT, WORK, -1, RWORK, -1, IERR ) 404 LWORK_TREVC = INT( WORK(1) ) 405 MAXWRK = MAX( MAXWRK, LWORK_TREVC ) 406 CALL ZHSEQR( 'S', 'V', N, 1, N, A, LDA, W, VR, LDVR, 407 $ WORK, -1, INFO ) 408 ELSE 409 IF( WNTSNN ) THEN 410 CALL ZHSEQR( 'E', 'N', N, 1, N, A, LDA, W, VR, LDVR, 411 $ WORK, -1, INFO ) 412 ELSE 413 CALL ZHSEQR( 'S', 'N', N, 1, N, A, LDA, W, VR, LDVR, 414 $ WORK, -1, INFO ) 415 END IF 416 END IF 417 HSWORK = INT( WORK(1) ) 418* 419 IF( ( .NOT.WANTVL ) .AND. ( .NOT.WANTVR ) ) THEN 420 MINWRK = 2*N 421 IF( .NOT.( WNTSNN .OR. WNTSNE ) ) 422 $ MINWRK = MAX( MINWRK, N*N + 2*N ) 423 MAXWRK = MAX( MAXWRK, HSWORK ) 424 IF( .NOT.( WNTSNN .OR. WNTSNE ) ) 425 $ MAXWRK = MAX( MAXWRK, N*N + 2*N ) 426 ELSE 427 MINWRK = 2*N 428 IF( .NOT.( WNTSNN .OR. WNTSNE ) ) 429 $ MINWRK = MAX( MINWRK, N*N + 2*N ) 430 MAXWRK = MAX( MAXWRK, HSWORK ) 431 MAXWRK = MAX( MAXWRK, N + ( N - 1 )*ILAENV( 1, 'ZUNGHR', 432 $ ' ', N, 1, N, -1 ) ) 433 IF( .NOT.( WNTSNN .OR. WNTSNE ) ) 434 $ MAXWRK = MAX( MAXWRK, N*N + 2*N ) 435 MAXWRK = MAX( MAXWRK, 2*N ) 436 END IF 437 MAXWRK = MAX( MAXWRK, MINWRK ) 438 END IF 439 WORK( 1 ) = MAXWRK 440* 441 IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY ) THEN 442 INFO = -20 443 END IF 444 END IF 445* 446 IF( INFO.NE.0 ) THEN 447 CALL XERBLA( 'ZGEEVX', -INFO ) 448 RETURN 449 ELSE IF( LQUERY ) THEN 450 RETURN 451 END IF 452* 453* Quick return if possible 454* 455 IF( N.EQ.0 ) 456 $ RETURN 457* 458* Get machine constants 459* 460 EPS = DLAMCH( 'P' ) 461 SMLNUM = DLAMCH( 'S' ) 462 BIGNUM = ONE / SMLNUM 463 CALL DLABAD( SMLNUM, BIGNUM ) 464 SMLNUM = SQRT( SMLNUM ) / EPS 465 BIGNUM = ONE / SMLNUM 466* 467* Scale A if max element outside range [SMLNUM,BIGNUM] 468* 469 ICOND = 0 470 ANRM = ZLANGE( 'M', N, N, A, LDA, DUM ) 471 SCALEA = .FALSE. 472 IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN 473 SCALEA = .TRUE. 474 CSCALE = SMLNUM 475 ELSE IF( ANRM.GT.BIGNUM ) THEN 476 SCALEA = .TRUE. 477 CSCALE = BIGNUM 478 END IF 479 IF( SCALEA ) 480 $ CALL ZLASCL( 'G', 0, 0, ANRM, CSCALE, N, N, A, LDA, IERR ) 481* 482* Balance the matrix and compute ABNRM 483* 484 CALL ZGEBAL( BALANC, N, A, LDA, ILO, IHI, SCALE, IERR ) 485 ABNRM = ZLANGE( '1', N, N, A, LDA, DUM ) 486 IF( SCALEA ) THEN 487 DUM( 1 ) = ABNRM 488 CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, 1, 1, DUM, 1, IERR ) 489 ABNRM = DUM( 1 ) 490 END IF 491* 492* Reduce to upper Hessenberg form 493* (CWorkspace: need 2*N, prefer N+N*NB) 494* (RWorkspace: none) 495* 496 ITAU = 1 497 IWRK = ITAU + N 498 CALL ZGEHRD( N, ILO, IHI, A, LDA, WORK( ITAU ), WORK( IWRK ), 499 $ LWORK-IWRK+1, IERR ) 500* 501 IF( WANTVL ) THEN 502* 503* Want left eigenvectors 504* Copy Householder vectors to VL 505* 506 SIDE = 'L' 507 CALL ZLACPY( 'L', N, N, A, LDA, VL, LDVL ) 508* 509* Generate unitary matrix in VL 510* (CWorkspace: need 2*N-1, prefer N+(N-1)*NB) 511* (RWorkspace: none) 512* 513 CALL ZUNGHR( N, ILO, IHI, VL, LDVL, WORK( ITAU ), WORK( IWRK ), 514 $ LWORK-IWRK+1, IERR ) 515* 516* Perform QR iteration, accumulating Schur vectors in VL 517* (CWorkspace: need 1, prefer HSWORK (see comments) ) 518* (RWorkspace: none) 519* 520 IWRK = ITAU 521 CALL ZHSEQR( 'S', 'V', N, ILO, IHI, A, LDA, W, VL, LDVL, 522 $ WORK( IWRK ), LWORK-IWRK+1, INFO ) 523* 524 IF( WANTVR ) THEN 525* 526* Want left and right eigenvectors 527* Copy Schur vectors to VR 528* 529 SIDE = 'B' 530 CALL ZLACPY( 'F', N, N, VL, LDVL, VR, LDVR ) 531 END IF 532* 533 ELSE IF( WANTVR ) THEN 534* 535* Want right eigenvectors 536* Copy Householder vectors to VR 537* 538 SIDE = 'R' 539 CALL ZLACPY( 'L', N, N, A, LDA, VR, LDVR ) 540* 541* Generate unitary matrix in VR 542* (CWorkspace: need 2*N-1, prefer N+(N-1)*NB) 543* (RWorkspace: none) 544* 545 CALL ZUNGHR( N, ILO, IHI, VR, LDVR, WORK( ITAU ), WORK( IWRK ), 546 $ LWORK-IWRK+1, IERR ) 547* 548* Perform QR iteration, accumulating Schur vectors in VR 549* (CWorkspace: need 1, prefer HSWORK (see comments) ) 550* (RWorkspace: none) 551* 552 IWRK = ITAU 553 CALL ZHSEQR( 'S', 'V', N, ILO, IHI, A, LDA, W, VR, LDVR, 554 $ WORK( IWRK ), LWORK-IWRK+1, INFO ) 555* 556 ELSE 557* 558* Compute eigenvalues only 559* If condition numbers desired, compute Schur form 560* 561 IF( WNTSNN ) THEN 562 JOB = 'E' 563 ELSE 564 JOB = 'S' 565 END IF 566* 567* (CWorkspace: need 1, prefer HSWORK (see comments) ) 568* (RWorkspace: none) 569* 570 IWRK = ITAU 571 CALL ZHSEQR( JOB, 'N', N, ILO, IHI, A, LDA, W, VR, LDVR, 572 $ WORK( IWRK ), LWORK-IWRK+1, INFO ) 573 END IF 574* 575* If INFO .NE. 0 from ZHSEQR, then quit 576* 577 IF( INFO.NE.0 ) 578 $ GO TO 50 579* 580 IF( WANTVL .OR. WANTVR ) THEN 581* 582* Compute left and/or right eigenvectors 583* (CWorkspace: need 2*N, prefer N + 2*N*NB) 584* (RWorkspace: need N) 585* 586 CALL ZTREVC3( SIDE, 'B', SELECT, N, A, LDA, VL, LDVL, VR, LDVR, 587 $ N, NOUT, WORK( IWRK ), LWORK-IWRK+1, 588 $ RWORK, N, IERR ) 589 END IF 590* 591* Compute condition numbers if desired 592* (CWorkspace: need N*N+2*N unless SENSE = 'E') 593* (RWorkspace: need 2*N unless SENSE = 'E') 594* 595 IF( .NOT.WNTSNN ) THEN 596 CALL ZTRSNA( SENSE, 'A', SELECT, N, A, LDA, VL, LDVL, VR, LDVR, 597 $ RCONDE, RCONDV, N, NOUT, WORK( IWRK ), N, RWORK, 598 $ ICOND ) 599 END IF 600* 601 IF( WANTVL ) THEN 602* 603* Undo balancing of left eigenvectors 604* 605 CALL ZGEBAK( BALANC, 'L', N, ILO, IHI, SCALE, N, VL, LDVL, 606 $ IERR ) 607* 608* Normalize left eigenvectors and make largest component real 609* 610 DO 20 I = 1, N 611 SCL = ONE / DZNRM2( N, VL( 1, I ), 1 ) 612 CALL ZDSCAL( N, SCL, VL( 1, I ), 1 ) 613 DO 10 K = 1, N 614 RWORK( K ) = DBLE( VL( K, I ) )**2 + 615 $ AIMAG( VL( K, I ) )**2 616 10 CONTINUE 617 K = IDAMAX( N, RWORK, 1 ) 618 TMP = CONJG( VL( K, I ) ) / SQRT( RWORK( K ) ) 619 CALL ZSCAL( N, TMP, VL( 1, I ), 1 ) 620 VL( K, I ) = DCMPLX( DBLE( VL( K, I ) ), ZERO ) 621 20 CONTINUE 622 END IF 623* 624 IF( WANTVR ) THEN 625* 626* Undo balancing of right eigenvectors 627* 628 CALL ZGEBAK( BALANC, 'R', N, ILO, IHI, SCALE, N, VR, LDVR, 629 $ IERR ) 630* 631* Normalize right eigenvectors and make largest component real 632* 633 DO 40 I = 1, N 634 SCL = ONE / DZNRM2( N, VR( 1, I ), 1 ) 635 CALL ZDSCAL( N, SCL, VR( 1, I ), 1 ) 636 DO 30 K = 1, N 637 RWORK( K ) = DBLE( VR( K, I ) )**2 + 638 $ AIMAG( VR( K, I ) )**2 639 30 CONTINUE 640 K = IDAMAX( N, RWORK, 1 ) 641 TMP = CONJG( VR( K, I ) ) / SQRT( RWORK( K ) ) 642 CALL ZSCAL( N, TMP, VR( 1, I ), 1 ) 643 VR( K, I ) = DCMPLX( DBLE( VR( K, I ) ), ZERO ) 644 40 CONTINUE 645 END IF 646* 647* Undo scaling if necessary 648* 649 50 CONTINUE 650 IF( SCALEA ) THEN 651 CALL ZLASCL( 'G', 0, 0, CSCALE, ANRM, N-INFO, 1, W( INFO+1 ), 652 $ MAX( N-INFO, 1 ), IERR ) 653 IF( INFO.EQ.0 ) THEN 654 IF( ( WNTSNV .OR. WNTSNB ) .AND. ICOND.EQ.0 ) 655 $ CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, N, 1, RCONDV, N, 656 $ IERR ) 657 ELSE 658 CALL ZLASCL( 'G', 0, 0, CSCALE, ANRM, ILO-1, 1, W, N, IERR ) 659 END IF 660 END IF 661* 662 WORK( 1 ) = MAXWRK 663 RETURN 664* 665* End of ZGEEVX 666* 667 END 668